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Results tagged with soft-question
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user 1946
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
4
votes
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
To my way of thinking, the arguments you mention seem not to distinguish sufficiently between the content of Goodstein's theorem as a universal claim $\forall p$ and Goodstein's theorem as a collectio …
- 236k
14
votes
Why is inner model theory evidence for consistency of large cardinals?
The explanation is philosophical rather than mathematical.
The idea is simply that the inner-model theory provides a rich account of what it would be like for the large cardinal axioms to be true, and …
- 236k
1
vote
Papers on history and philosophy of mathematics suitable for master's students
For the philosophy of mathematics, I wrote my book specifically with mathematical readers in mind. Many readers have told me that they appreciate the accessible manner the book has of treating even su …
10
votes
Accepted
Why include $0$ and $1$ in the signature of Presburger arithmetic?
It is the same in Peano arithmetic, where the standard language is $\{+,\cdot,0,1,<\}$ for the standard model $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, even though $0$, $1$, and $<$ are definable from …
- 236k
1
vote
Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another example.
The maximality principle in forcing is the scheme asserting of every statement $\varphi$ in the language of set theory that if there is forcing extension $V[G]$ of the set-the …
- 236k
6
votes
Accepted
Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements.
Joel David Hamkins and Bokai Yao, Reflection in se …
- 236k
5
votes
Decision problems for which it is unknown whether they are decidable
It remains open whether the won-position problem of infinite chess is decidable, the problem of determining whether a given finite position in infinite chess is winning for white or not. See Richard S …
21
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …
26
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …
35
votes
What programming language should a professional mathematician know?
My answer is: TikZ
This is a programming language, often used in combination with
LaTeX, for producing high-quality graphics.
I view this language as important for mathematicians, not because
mathem …
61
votes
Naming in math: from red herrings to very long names
Let me mention as a counterpoint that there is less need for
new terminology than one might expect. Mathematical exposition
is often more successful and clearer without new terminology, and
one should …
86
votes
Has incorrect notation ever led to a mistaken proof?
Here is an example from set theory.
Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory o …
4
votes
Mathematical objects whose name is a single letter
In set theory, $V$ is universally known as the universe of all sets.
Similarly, meanwhile, $L$ is the constructible universe.
6
votes
Mathematical games interesting to both you and a 5+-year-old child
I often play the game Doubled, Squared, Cubed! with my kids, as I did as I child myself years ago with my siblings. It can be played with kids of any age, and it is a great way to expose the kids to n …
36
votes
Accepted
Dealing with unwanted co-authorship requests
Well, of course the young mathematician should simply discuss the
matter with the senior mathematician and perhaps the student until
they can come to an agreeable arrangement. My advice is that they
s …